Magnetic current

Magnetic displacement current

Magnetic displacement current or more properly the magnetic displacement current density is the familiar term ∂B/∂t. It is one component of M t .

M t = ∂ B ∂ t + M i .

where

M t = total magnetic current. M i = impressed magnetic current (energy source).

Electric vector potential

The electric vector potential, F, is computed from the magnetic current density, M i , in the same way that the magnetic vector potential, A, is computed from the electric current density. F is used in the same way the magnetic vector potential for sources attributed to magnetic current. Examples of use include finite diameter wire antennas and transformers.

magnetic vector potential: A ( r , t ) = μ 0 4 π ∫ Ω J ( r ′ , t ′ ) | r − r ′ | d 3 r ′ .

electric vector potential: F ( r , t ) = ϵ 0 4 π ∫ Ω M i ( r ′ , t ′ ) | r − r ′ | d 3 r ′ .

where F at point r and time t is calculated from magnetic currents at distant position r ′ at an earlier time t ′ . The location r ′ is a source point within volume Ω that contains the magnetic current distribution. The integration variable, d 3 r ′ , is a volume element around position r ′ . The earlier time t ′ is called the retarded time, and calculated as

t ′ = t − | r − r ′ | c .

Retarded time accounts for the accounts for the time required for electromagnetic effects to propagate from point r ′ to point r .

Phasor form

When all the functions of time are sinusoids of the same frequency, the time domain equation can be replaced with a frequency domain equation. Retarded time is replaced with a phase term.

F ( r ) = ϵ 0 4 π ∫ Ω M i ( r ) e ( − j k | r − r ′ | ) | r − r ′ | d 3 r ′ .

where F and M i are phasor quantities and k is the wave number.

Magnetic frill generator

A distribution of magnetic current, commonly called a magnetic frill generator, may be used to replace the driving source and feed line in the analysis of a finite diameter dipole antenna. The voltage source and feed line impedance are subsumed into the magnetic current density. In this case, the magnetic current density is concentrated in a two dimensional surface so the units of M i are volts per meter.

The inner radius of the frill is the same as the radius of the dipole. The outer radius is chosen so that

Z L = Z 0 ln ⁡ ( b a ) .

where

Z L = impedance of the feed transmission line (not shown). Z 0 = impedance of free space.

The equation is the same as the equation for the impedance of a coaxial cable. However, a coaxial cable feed line is not assumed and not required.

The amplitude of the magnetic current density phasor is given by:

M i = k ρ with a ≤ ρ ≤ b .

where

ρ = radial distance from the axis. k = V s ln ⁡ ( b a ) . V s = magnitude of the source voltage phasor driving the feed line.